where the QD level position is fixed to \(\epsilon _0=-U/2\)  to describe the spinful odd CB valley.
This equation is the expression for YSR bound states [1]} written in the language of our system and including the external magnetic flux (through both the LP modulation of the superconducting gap \(\Delta (\phi )\)  and the Zeeman effect \(V_Z\) ). The dashed green curves in Fig.  REFD are calculated with this analytical expression without fitting parameters (since the energy of the ABSs at zero magnetic field fixes \(\Gamma _S\) ). The excellent agreement between Eq. (REF ) and the experiment demonstrate that, indeed, our ABSs are YSR singlets (for comparison, the excitations to the BCS-like singlet \(\mathinner {|{S}\rangle }=\mathinner {|{0}\rangle }- \mathinner {|{\uparrow \downarrow }\rangle }\)  would occur at a much higher energy, of order \(\zeta \approx \pm (U-\Gamma _S)/2=0.85\)  meV\(\gg \Delta \) ).
